Cartier isomorphism and Hodge Theory in the non-commutative case

These are lecture notes from Clay Summer School in Goettingen, in 2006; the lectures were an attempt at an elementary introduction to math.KT/0611623.

💡 Research Summary

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The lecture notes present a systematic extension of the classical Cartier isomorphism to the realm of non‑commutative differential graded (DG) algebras and use this extension to prove a degeneration theorem for the Hodge‑to‑de Rham spectral sequence in the non‑commutative setting. The exposition begins with a concise review of the commutative case: over a perfect field k of characteristic p > 0, the Cartier isomorphism identifies the Frobenius‑twisted de Rham complex Ω⁎_{X^{(1)}} with the de Rham complex Ω⁎_X, and this identification forces the Hodge‑to‑de Rham spectral sequence of a smooth proper variety X to collapse at E₁.

The notes then shift to the algebraic side, introducing the basic homological invariants of a DG‑algebra A: Hochschild homology HHₙ(A), cyclic homology HCₙ(A), and Connes’ B‑operator. The authors recall the standard long exact sequence linking Hochschild, cyclic, and periodic cyclic homology and explain how the natural filtration on HCₙ(A) yields a Hodge‑type spectral sequence whose E₁‑page is HHₙ(A) with a formal variable u of degree 2.

The core of the work is the construction of a “non‑commutative Cartier map”. Assuming that A is smooth, proper, and liftable to the second Witt vectors W₂(k), the authors define a Frobenius twist A^{(1)} and a pull‑back map on Hochschild chains induced by the Frobenius endomorphism of k. They then analyze the interaction between the Bockstein differential arising from reduction modulo p and Connes’ B‑operator, establishing the key identity β = B ∘ F^{−1} (up to the usual sign conventions). This identity yields an explicit isomorphism \